Statistics and Probability Answers

Obtain the mean and variance of the discrete random variable X having the probability
density function

f (x) = 2x for 0 ≤ x ≤ 1 and

0 otherwise.

The number of revolutions per minute (x) and power (y) hp of a diesel engine are tabulated
below.

x 400 500 600 700 750
y 580 1030 1420 1880 2100

Obtain the regression equation for this data.

The sales per day in a shop is exponentially distributed with average sale amounting to Rs.100 and net profit is 8%. Find the probability that the net profit exceeds Rs.30 on two consecutive days

Amir, Waseem, Afridi, and Akhtar play the following game. Each man takes one of four balls numbered I though 4 from a jug. The man who draws ball 4 loses. The other three return their balls to the jug and draw again. Now the one who draws ball 3 loses. The other two return their balls to the jug and draw again. The man who draws ball 1 wins the game.
i) What is the probability that Akhtar does not lose in the first two draws?
ii) What is the probability that Afridi wins the game?

The sales per day in a shop is exponentially distributed with average sale amounting to Rs.100 and net profit is 8%. Find the probability that the net profit exceeds Rs.30 on two consecutive days

A stamping machine produces ‘can tops’ whose diameters are normally distributed with a standard deviation of 0.01 inch. At what nominal mean diameter should the machine be set, so that no more than 5 % of the ‘can tops’ produced have diameters exceeding 3 inches?

i) Suppose that the life of a certain type of electron tube has an exponential distribution with a mean life of 500 hours. Find the probability that it will last for another 600 hours if the tube has been in operation for 300 hours.
ii) For a random variable X , E(X)=10 , and Var(X)=25. Find the positive values of a and b such that Y=ax-b has expectation zero and variance 1.

The probability that a new airport will get an award for its design is 0.16, the
probability that it will get
an award for the efficient use of materials is 0.24 and the
probability that it will get both the awards is 0.11.
(a)
what is the probability that it will get atleast one of the two awards?
(b)
what is the probability that it will get only one of the two awards?

In a test on electric bulbs, it was found that the life time of a
particular brand was
normally distributed with an average life of 2000 hours and S.D. of 60 hours. If a
firm purchases 2500 bulbs, find the number of bulbs that are likely to last for (i)
more than 2100 hours, (ii) less than 1950 hours and (iii) betwe
en 1900 and 2100
hours.

The joint probability function of two discrete random variables X and Y is given by
f(x
,y) = c(2x+y)
, where
x
and
y
can assume all integral values such that
0
≤
x
≤
2,
0
≤
y
≤
3
, and
f(x, y) = 0
otherwise. Find the value of th
e constant
c
,
(ii) P(x=2,
y=1), (iii) P(x≥1, y≤2)
, (iv) Marginal probability functions of X and Y. Check
whether X and Y are independent